Complex analytic variety

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In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety [note 1] or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions.


Denote the constant sheaf on a topological space with value by . A -space is a locally ringed space , whose structure sheaf is an algebra over .

Choose an open subset of some complex affine space , and fix finitely many holomorphic functions in . Let be the common vanishing locus of these holomorphic functions, that is, . Define a sheaf of rings on by letting be the restriction to of , where is the sheaf of holomorphic functions on . Then the locally ringed -space is a local model space.

A complex analytic variety is a locally ringed -space which is locally isomorphic to a local model space.

Morphisms of complex analytic varieties are defined to be morphisms of the underlying locally ringed spaces, they are also called holomorphic maps. A structure sheaf may have nilpotent element,[1] and also, when the complex analytic space whose structure sheaf is reduced, then the complex analytic space is reduced, that is, the complex analytic space may not be reduced.

An associated complex analytic space (variety) is such that;[1]

Let X be schemes finite type over , and cover X with open affine subset () (Spectrum of a ring). Then each is an algebra of finite type over , and . Where are polynomial in , which can be regarded as a holomorphic function on . Therefore, their common zero of the set is the complex analytic subspace . Here, scheme X obtained by glueing the data of the set , and then the same data can be used to glueing the complex analytic space into an complex analytic space , so we call a associated complex analytic space with X. The complex analytic space X is reduced if and only if the associated complex analytic space reduced.[2]

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  1. ^ a b Hartshorne 1977, p. 439.
  2. ^ Grothendieck & Raynaud (2002) (SGA 1 §XII. Proposition 2.1.)


  1. ^ Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced


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